Playbook
Complex Numbers
43 q - 1.00/paper - 33% HARD. Three pages in notes order. Modulus and Argument (17 q, 29% HARD) is the half worth owning; Algebra with the cube roots of unity (15 q, 47%) is the expensive corner; Locus (11 q, 18%) is one geometric idea - a modulus is a distance.
- Questions in the bank
- 43
- q/paper (2024–25 shifts)
- 1.00
- Tagged HARD
- 33%
- Subtopics
- 3
Strand: Long Tail
When you’ll see it
An i in the expression — a modulus or argument asked for, a cube root of unity, or a locus described by a modulus condition.
How this chapter is tested
43 q, 1.00/paper, 33% HARD, and it splits about as cleanly as any chapter in the bank. Modulus and Argument is 17 q at 29% HARD and Locus 12 q at 17% — the softest pages anywhere in the long tail. Algebra with the cube roots of unity is 15 q at 47%. Every PYQ is tagged to one of the three notes pages at /notes/mht-cet-maths/complex-numbers.
That asymmetry IS the strategy. Own the modulus and locus pages and treat the algebra page as opportunistic: under 30% HARD at one question a paper is about as close to free marks as the tail offers, and it is reachable with the identities of Trigonometry - I plus De Moivre's theorem.
The harder page runs largely on omega and on a polynomial evaluated at a complex x. Three facts about the cube roots of unity answer the omega stems: omega cubed is 1, 1 + omega + omega squared is 0, and powers of omega cycle with period 3. The polynomial stems are answered by the minimal quadratic of the given root, never by direct substitution.
Locus questions are circles and lines in disguise — a condition of the form |z - a| = r is a circle of radius r centred at a. That is also where the cross-chapter extremum lives: the greatest and least modulus of z on such a disc is |a| plus or minus r, exactly the Circle chapter's distance-to-centre move, with no calculus and no differentiation of a modulus.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Algebra of Complex Numbers — Conjugates, Powers of i and Cube Roots of Unity
Reduce powers of i modulo 4, rationalise by the conjugate, equate real and imaginary parts, evaluate a polynomial at a complex root via its minimal quadratic, and reduce powers of omega modulo 3.
Modulus and Argument — Polar Form, De Moivre and Square Roots
Moduli multiply and divide, so never expand for a modulus. The argument comes from the ratio of the parts AND the quadrant, never the ratio alone. z = r(cos theta + i sin theta) makes powers routine; |z| + z = a + ib has a closed form.
Locus in the Argand Plane — Circles, Lines and Greatest/Least Modulus
|z - a| = r is a circle; |z - a| = |z - b| is the perpendicular bisector; a purely-imaginary quotient is a circle after rationalising; greatest and least |z| on a disc are |a| + r and |a| - r.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Argument from the ratio alone
The arctangent of the ratio gives a reference angle. Points in the second and third quadrants need pi added or subtracted, and the un-adjusted angle is always an option.
Modulus distributed over a sum
|z1 z2| = |z1| |z2| is true; |z1 + z2| = |z1| + |z2| is not, except in a degenerate case. The additive version is a planted distractor.
Principal argument out of range
The principal argument lies in (-pi, pi]. A value outside that interval must be shifted by 2 pi, and the unshifted value is offered.
Unreduced powers of omega
omega^4 is omega and omega^5 is omega squared. Leaving a high power unreduced produces an expression that looks unlike any option, which usually means the reduction was skipped.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Complex Numbers notesDrill every complex numbers question
43 questions from the bank, scoped to 3 bundled subtopics.
Drill one subtopic at a time
The 3 subtopics this playbook covers, in catalog order.
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